The Elliptic Gamma Function and SL(3, Z)⋉Z3

Felder, Giovanni; Varchenko, Alexander

doi:10.1006/aima.2000.1951

Cited by 131 publications

(295 citation statements)

References 8 publications

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“…As shown in [21], SL(3, Z) transformations are related to these matching conditions, since they reproduce all anomaly coefficients through the cocycle phase factor emerging in the corresponding transformation rule for the elliptic gamma function [22].…”

Section: Jhep07(2017)041mentioning

confidence: 92%

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Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Brünner

Regalado

Spiridonov

2017

J. High Energ. Phys.

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We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3, Z) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N = 2 SCFT with E 6 flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N = 1 theory with enhanced E 7 flavour symmetry.

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Section: Jhep07(2017)041mentioning

confidence: 92%

“…For |p|, |q| < 1 it is related to the standard elliptic gamma function by G(u; ω) = Γ(re −2πiu/ω 1 ;q, r)Γ(e 2πiu/ω 2 ; p, q), (2.5) withq = exp(−2πiω 2 /ω 1 ). As follows from an identity derived in [22], the modified elliptic gamma function can be rewritten as 6) JHEP07(2017)041…”

Section: Jhep07(2017)041mentioning

confidence: 99%

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Brünner

Regalado

Spiridonov

2017

J. High Energ. Phys.

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“…It is easy to check [38], that the function G(u; ω) = e − πi 3 B3,3(u|ω) Γ(e −2πi u ω 3 ;r,p), (2.21) where |p|, |r| < 1, satisfies the same three equations and the normalization as the function (2.20). Therefore these functions coincide, and their equality constitutes one of the laws of the modular transformations for the elliptic gamma function related to the SL(3; Z)-group [25]. From the expression (2.21) it follows, that G(u; ω) is a meromorphic function of u for ω 1 /ω 2 > 0, i.e.…”

Section: The Elliptic Gamma Functionsmentioning

confidence: 94%

“…through which the elliptic gamma function appeared implicitly in the work of Baxter on the eight-vertex model [6] (see also [5,25]). …”

Section: The Elliptic Gamma Functionsmentioning

confidence: 99%

“…Importance of the elliptic gamma function was stressed by Ruijsenaars [18], who gave to it this name and considered some of its properties. Modular transformations of this function were described in [25]. In [11], a modified elliptic gamma function was built which remains also well defined in the case when one of the base parameters lies on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

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Essays on the theory of elliptic hypergeometric functions

Spiridonov¹

2008

Russ. Math. Surv.

118

177

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Abstract. We give a brief review of the main results of the theory of elliptic hypergeometric functions -a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

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Q-Deformed KZB Heat Equation: Completeness, Modular Properties and SL(3, Z)

Felder¹,

Varchenko

2002

Advances in Mathematics

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We study the properties of one-dimensional hypergeometric integral solutions of the q-difference (''quantum'') analogue of the Knizhnik-Zamolodchikov-Bernard equations on tori. We show that they also obey a difference KZB heat equation in the modular parameter, give formulae for modular transformations, and prove a completeness result, by showing that the associated Fourier transform is invertible. These results are based on SLð3; ZÞ transformation properties parallel to those of elliptic gamma functions. # 2002 Elsevier Science (USA) INTRODUCTIONIn this paper we continue the study of the q-analogue of the KnizhnikZamolodchikov-Bernard (qKZB) equations on elliptic curves and their solutions initiated in [FTV1, FTV2, FV1].In [FTV1], hypergeometric solutions of qKZB equations were introduced. In [FTV2], the monodromy of hypergeometric solutions was calculated, and a symmetry between equations and monodromies was discovered: the equations giving the monodromy are again qKZB equations but with modular parameter and step of the difference equations exchanged. In [FV1], we introduced the q-analogue of the KZB heat equation, which 1 To whom correspondence should be addressed.

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The Elliptic Gamma Function and SL(3, Z)⋉Z3

Cited by 131 publications

References 8 publications

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Supersymmetric Casimir energy and S L 3 ℤ $$ \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) $$ transformations

Essays on the theory of elliptic hypergeometric functions

Q-Deformed KZB Heat Equation: Completeness, Modular Properties and SL(3, Z)

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